What purpose does the webbing serve? Is it a structural thing for a better base or accuracy or something?

Not sure on that, I do know every 'thumbhole' style stock I've held/fired has altered my grip.

Hard to describe, but the weight of my hand rest on a different part of my hand, which helps with trigger squeeze. Basically, with the thumbhole type, I'm resting on the bit of web, where on the plain pistol grip type, or on the normal rifle stock, I actually have to use the muscles in my hand to hold onto the stock.
For me anyway, on a good stock. Not all stocks, even of the same style, are equal of course.

well, i know that with the prone position grip that i use, (left arm on ground, bent around the grip) the left forearm would be against that webbing, adding an extra layer of recoil control;

but as an engineering student, i look at the stock of the rifle:
a large part of accurate rifle design focuses on how the recoil is taken by the user. the general consensus is that the more that the recoil is taken linearly, the straighter the round will travel.

to achieve that goal, the designers tried to make the grip as high as they could without making the rifle structurally weak. so, that in turn would explain the webbing. without the webbing there, if you dropped the rifle repeatedly, or just due to plain old enviornmental issues, the weakest point in the rifle (the small base of the stock) would snap from the lack of being able to handle any moment/shear stress about that point, or damage from warpiing. and so, from a structural standpoint, that webbing makes a huge impact on the durability and accurate use of the rifle. many tactical sniper rifles incorporate the design elements i stated.
also, i'm sure the webbing creates a specific type of grip.

for all the CS players who live in la-la land, here's a tidbit:
i looked through my copy of 'Jane's Security and CO-IN Equipment' (i bought it at a yard sale, shut up.) and found the most famous example (the gun or replicas in the pictures above), the so-called AWP. in the game, it shoots a .338 lupua magnum, and is commonly refered to as the AWM by computer game players as well, the 'artic warfare magnum'. however, both names are wrong. the specific gun in question is actually made by an american company named 'Accuracy International' and the gun is actually called the '.338 Super Magnum sniper rifle.' the people who made CS fucked up, imho, b/c the same company makes the version called the '7.62 Counter Terrorist sniping rifle' not any more tongue-in-cheek than that.

i figured i'd add in that little paragraph b/c not everyone in this subforum shoots real guns, apparantly. w/e.

well, i know that with the prone position grip that i use, (left arm on ground, bent around the grip) the left forearm would be against that webbing, adding an extra layer of recoil control;

but as an engineering student, i look at the stock of the rifle:
a large part of accurate rifle design focuses on how the recoil is taken by the user. the general consensus is that the more that the recoil is taken linearly, the straighter the round will travel.

to achieve that goal, the designers tried to make the grip as high as they could without making the rifle structurally weak. so, that in turn would explain the webbing. without the webbing there, if you dropped the rifle repeatedly, or just due to plain old enviornmental issues, the weakest point in the rifle (the small base of the stock) would snap from the lack of being able to handle any moment/shear stress about that point, or damage from warpiing. and so, from a structural standpoint, that webbing makes a huge impact on the durability and accurate use of the rifle. many tactical sniper rifles incorporate the design elements i stated.
also, i'm sure the webbing creates a specific type of grip.

for all the CS players who live in la-la land, here's a tidbit:
i looked through my copy of 'Jane's Security and CO-IN Equipment' (i bought it at a yard sale, shut up.) and found the most famous example (the gun or replicas in the pictures above), the so-called AWP. in the game, it shoots a .338 lupua magnum, and is commonly refered to as the AWM by computer game players as well, the 'artic warfare magnum'. however, both names are wrong. the specific gun in question is actually made by an american company named 'Accuracy International' and the gun is actually called the '.338 Super Magnum sniper rifle.' the people who made CS fucked up, imho, b/c the same company makes the version called the '7.62 Counter Terrorist sniping rifle' not any more tongue-in-cheek than that.

i figured i'd add in that little paragraph b/c not everyone in this subforum shoots real guns, apparantly. w/e.

That's really informational and helpful. My friend is carving a buttstock for fun and we were wondering what that webbing was for as far as mechanics go. Aesthetically it's fucking awesome looking.

I took a guess at it being something structural and better for absorbing recoil. I guess I was partly right.

Edit:

I do shoot real guns but I live in california and our choices are limited and it's expensive to shoot here ($15 for a box of 40 9mm rounds for a Glock, Even with the discount from being buddies with range people). And I'm a poor college student.*

I believe the webbing you are describing initially started to show up after Assault Rifle legislation became popular. One of the determining characteristics ,among others, was a pistol grip. The thumbhole/webbing gave the same degree of control as a pistol grip on a rifle allowing manufacturers to remove one of the legal definitions of a assault rifle. Look up Roos-Roberti assault gun legislation under California penal codes for more info. The design characteristic looks like it has filtered down to bolt action rifles in some cases.

honesty, a lot of that kind of **** is marketing and not engineering per se. "Because it looks cool." is the answer a lot of the time.

well, looking at the stock, it was more of a forced option rather than a 'cool design' thing. because without the webbing, it's just a little ridiculous to have a stock base that thin just to accomodate for a high grip on the design of the rifle, hence the same design on many rifles/shotguns using the same idea. also, joe is correct, the grip forces a 'pistol grip,' for added stability, also keeping with the idea of having a higher grip, and taking more force straight on.

Thumbhole stocks go way back. On target guns, they're a good way to keep weight in the gun, and some people just like the aesthetics. Also, if your rifle is not designed to have a separate pistol grip attached (as an AR-15 or an AK-47 is) then there is some strength and stiffness advantage to the thumbhole. And as vsc87 says, a high grip is a good grip on a gun. On a rifle, the point is to put the boreline of the rifle (the path the bullet travels) as low as you can so it pushes straight back into the shooter's shoulder. If the boreline is higher, then recoil pushes the top of the stock back but the bottom is stopped by the shoulder, so the gun tries to rotate upward. If the recoil impulse can travel straight back, muzzle climb is less pronounced.

Guns like the AR-15/M16 family were designed to use that principle without modifications. That's why the stock is inline with the bore, and that in turn is why they have to have pistol grips--the stock is too high for a "conventional grip" carved into the stock to be very practical.
The reason guns designed for separate pistol grips began to be sold with thumbhole stocks (usually ugly, uncomfortable thumbhole stocks) was to reduce the number of "evil features" under the federal Assault Weapons Ban. The people who wrote it had no idea what an "assault weapon" was, but they knew it looked scary and military, so they made up a list of features they thought "hunting guns" wouldn't have in the hopes that they could ban everything else that way. Tragically, they were idiots and thus unable to distinguish between functional differences and aesthetic differences. Thus, their list of Evil Features went something like this:

1. Pistol grip ("grip that protrudes significantly and is separate from the buttstock.")
2. Bayonet Lug (This is the small piece of metal to which the lock on a bayonet attaches. Without it, you can't lock your bayonet in place and so drive-by bayonettings are prevented.)
3. Ability to accept detachable magazines holding more than ten rounds. (Because if you can't rob a bank with ten rounds or less, you don't deserve the money anyway.)
4. Flash suppressor. The flash suppressor diverts burning gases out in multiple directions perpendicular to the barrel so you don't get a big fireball in front of you when you fire. During the day that's no big deal, but at night, that fireball can be blinding, and the shorter the barrel the worse it gets. The flash suppressor preserves your night vision.
That makes it too deadly for civlians to own; besides, why would a civilian be out shooting at night, anyway? Goddamn citizens run around like they own the place.

Under the feds, you could have, I think, two of these features. So on an AR-15, most manufacturers figured there was no need to change the stock, and they weren't about to make a gun that couldn't accept 99% of the magazines on the market, so they eliminated flash suppressors, bayonet lugs. That was the entire difference between a pre-ban and a post-ban AR-15.
On the AK-47 clones, the thumbhole stocks became popular. Don't ask me why; the most common kind were Chinese MAK-90's and the thumbhole version was the ugliest gun ever produced outside a Nambu factory.

There are still states where pre-ban and post-ban mean something, but I'm not up on the details.